Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 133

Proof. Suppose the conditions on a and b are satisfied. We first show that for every
pair of functionals (f, g) ∈ X × M , there is exactly one solution (u, λ) of the
saddle point problem satisfying

+ β − 1 1 g ,

C
α

u ≤ α − 1

f +

( 4 . 12 )

β − 1 1 β − 1 1

C
β

C
α

C
α

λ ≤ + f + + g .

V(g) is not empty for g ∈ M . Indeed, by Lemma 4.2(ii), there exists u 0 ∈ V ⊥
with

Bu 0 = g.

u 0 ≤ β − 1

Moreover, g .

With w : = u − u 0 , (4.4) is equivalent to

a(w, v) + b(v, λ) = f, v − a(u 0 ,v) for all v ∈ X,

( 4 . 13 )

b(w, µ) = 0 for all µ ∈ M.

By the V -ellipticity of a , the function

1
2 a(v, v)

− f, v + a(u 0 ,v)

attains its minimum for some w ∈ V with

w ≤ α − 1 ( f + C u 0 ) ≤ α − 1 ( f + Cβ − 1 g ).

In particular, the Characterization Theorem II.2.2 implies

a(w, v) = f, v − a(u 0 ,v) for all v ∈ V. ( 4 . 14 )

The equations (4.13) will be satisfied if we can find λ ∈ M such that

b(v, λ) = f, v − a(u 0 + w, v) for all v ∈ X.

The right-hand side defines a functional in X , which in view of (4.14) lies in V 0 .
By Lemma 4.2(iii), this functional can be represented as B λ with λ ∈ M , and

λ ≤ β − 1 ( f + C u ).

This establishes the solvability. The inequalities (4.12) follow from the bounds on
u 0

, w , and λ and the triangle inequality u ≤ u 0 + w .